Gauss' method to solve the following system of equations

Question

I need to use Gauss' method to solve the following system of equations and to describe its solution set. Can anyone help me getting started. \begin{alignat*}{8} x & + & y & + & z & - & w & = & 1\\ & & y & - &z & + & w & = & -1\\ 3x & & & + & 6z & - & 6w & = & 6\\ & - & y & + & z & - & w & = & 1 \end{alignat*}

Answer 1

The Gauss method is based on eliminating several variables in the system of linear equations by taking linear combinations. For example:

$1*(x+y+z-w)+1*(y-z+w)=x+2y=1*1+1*(-1)=0$ (added first equation to second).

$1*(3x+6z-6w)+(-6)*(-y+z-w)=3x+6y=1*6+(-6)*1=0$ (linear combination of third equation and 4th equation)

Now you have new linear equations $(A) x+2y=0, (B) 3x+6y=0$. Again you can take linear combinations like this: $3*(A)+(-1)*(B)=3*0+(-1)*0=0$. The linear System defined by (A) and (B) has no unique solution; therefore you can consider $x$ as a Parameter and you have: $y=- \frac{x}{2}$.